Following [1]} (see also [2]}) we call a set \(\mathcal {A}\)  of graphs smooth if \(|\mathcal {A}_n|/(n |\mathcal {A}_{n-1}|)\)  tends to a limit as \(n \rightarrow \infty \)  (which must the growth constant \(\gamma _\mathcal {A}\) ).
Bender, Canfield and Richmond [3]} showed that the class \(\mathcal {E}^S\)  of graphs embeddable in a given surface \(S\)  is smooth, and this also follows from the asymptotic formula for \(|\mathcal {E}^S_n|\)  in [4]}.
In contrast, the graph classes considered above
(for example in Theorem REF )
need not be smooth.
(In Theorem REF  we could say take \(\mathcal {A}\)  as the class of graphs \(G\)  which are embeddable in the sphere \(\mathbf {S}_0\)  if \(v(G)\)  is even and in the torus \(\mathbf {S}_1\)  if \(v(G)\)  is odd. Observe that in this case the class of connected graphs which are free for \(\mathcal {A}\)  is the class of connected planar graphs \(G\)  with \(v(G)\)  even.)
